26.1 Introduction to Differential Equations

A differential equation is an equation involving derivatives. The
order of the equation is the highest derivative occurring in the equation.

Here are some examples

The first four of these are first order differential equations,
the last is a second order equation.

The first two are called linear differential equations because
they are linear in the variable y, the first has an "inhomogeneous
term" that is independent of y on the right, the second is a homogeneous
linear equation since all terms are linear in y.

The first three of these are "separable" differential
equations, since they can be rewritten as dx f(x) = dy g(y) for appropriate
f and g.

If you know only the derivative of a function, you do not have enough information
to determine it completely. You can therefore seek either a solution
to a differential equation, or a general solution (which usually has
a constant for each order of the equation in it) or a solution subject to
some additional condition or conditions.

You can find the general solution to any separable first order differential
equation by integration, (or as it is sometimes referred to, by "quadrature").
All you need do is to integrate both sides of the equation dx f(x) = dy g(y).
Thus you can apply the numerical techniques of the previous chapter to each
of these directly and solve them numerically, if you cannot integrate them exactly.

The question we address here is:

Suppose we have a first order differential equation that is not separable,
so we cannot reduce its solution to quadratures directly. Can we apply the numerical
techniques previously for doing integrals to the task of solving these equations?

The answer is yes and we show how below. There is indeed a complication which
we discuss next, but it can be overcome.

The implication of this fact is, that any system whose behavior can be modeled
by a first order differential equation, or even by a set of linked first order
equations, can be solved numerically to any desired accuracy by a modern computer
very quickly. This makes possible real time control of such systems and is of
great value in engineering.